Question

Evaluate \(f(5)\) if
\[f(x)-x\,f(-x)=x^2\]

Answer 1

Please note thatif I make this substitution: x--->-x, I can re-write your equation as below:
\[f(-x)+xf(x)=x ^{2}\]
now I multiply both sides by x, and I get:
\[xf(-x)+x ^{2}f(x)=x ^{3}\]
adding, side by side, that last above equation and your equation, I get this equation:
\[(x ^{2}+1)f(x)=x ^{3}+x ^{2}\]
and:
\[f(x)=\frac{ x ^{3}+x ^{2} }{ x ^{2}+1 }\]
so substituting x=5, I get:
\[f(5)=\frac{ 125+25 }{ 25+1 }=\frac{ 150 }{ 26 }=\frac{ 75 }{ 13 }\]

Answer 2

That's correct! Nice work

Answer 3

Thanks! @SithsAndGiggles

Answer 4

I saw this listed in a math contest from a while back and it looked like an interesting problem :)

Answer 5

ok!

Answer 6

nice question! :)